Part 1: Neighborhood environments in New York City

Question 1: Run an appropriate unsupervised learning analysis that will address the above research question. Your analysis should include a data-driven approach to determine the optimal number of outputs that are retained/produced by the chosen learning technique.

First, I read in the nyc environment dataset and removed any of the NAs in the dataset (there were no missing data in this data set). I then checked if scaling was needed by looking at the means and standard devations. Since the means and SDs were similar to each other I did not think that it was necessary to scale the data.

nyc.env<-read.csv("nyc_env_correct.csv", header=TRUE)

nyc.env.nomiss<-na.omit(nyc.env)

#Check means and SDs to determine if scaling is necessary
colMeans(nyc.env.nomiss, na.rm=TRUE)
##                Pct.Low.Inc                Diesel.Conc 
##                  0.3663891                  3.9260364 
## NPL.Site.Count..wt.by.dist        Pct.pre1960.Housing 
##                  0.1573208                  0.6720128
apply(nyc.env.nomiss, 2, sd, na.rm=TRUE)
##                Pct.Low.Inc                Diesel.Conc 
##                  0.1932175                  1.5077004 
## NPL.Site.Count..wt.by.dist        Pct.pre1960.Housing 
##                  0.1724710                  0.2312597

A clustering analysis was done next using k-means clustering to assess if there are any clusters in the data and where they are. The gap statistic determined that 8 clusters were needed for this data and that was entered to the final cluster model.

set.seed(100)
clusters<-kmeans(nyc.env, 4, nstart=25)
str(clusters)
## List of 9
##  $ cluster     : int [1:2140] 1 2 2 2 1 3 1 2 1 4 ...
##  $ centers     : num [1:4, 1:4] 0.383 0.302 0.355 0.406 2.22 ...
##   ..- attr(*, "dimnames")=List of 2
##   .. ..$ : chr [1:4] "1" "2" "3" "4"
##   .. ..$ : chr [1:4] "Pct.Low.Inc" "Diesel.Conc" "NPL.Site.Count..wt.by.dist" "Pct.pre1960.Housing"
##  $ totss       : num 5120
##  $ withinss    : num [1:4] 59.1 54.6 73.2 115.7
##  $ tot.withinss: num 303
##  $ betweenss   : num 4818
##  $ size        : int [1:4] 307 343 815 675
##  $ iter        : int 2
##  $ ifault      : int 0
##  - attr(*, "class")= chr "kmeans"
clusters
## K-means clustering with 4 clusters of sizes 307, 343, 815, 675
## 
## Cluster means:
##   Pct.Low.Inc Diesel.Conc NPL.Site.Count..wt.by.dist Pct.pre1960.Housing
## 1   0.3826832    2.219956                  0.1859193           0.4013800
## 2   0.3017552    1.324008                  0.1125524           0.3606563
## 3   0.3547306    4.568935                  0.1321465           0.8047739
## 4   0.4058986    5.247962                  0.1974583           0.7930192
## 
## Clustering vector:
##    [1] 1 2 2 2 1 3 1 2 1 4 1 1 4 1 4 1 1 1 3 1 1 1 3 1 3 3 4 4 4 1 4 3 4 3
##   [35] 1 3 4 1 4 4 1 1 3 1 3 4 2 1 3 1 3 1 4 2 4 3 1 4 3 3 1 3 3 2 1 1 2 1
##   [69] 1 1 1 1 1 1 2 1 2 4 4 2 1 4 2 4 4 4 4 4 3 1 4 1 2 1 2 4 3 3 1 3 1 1
##  [103] 1 4 4 4 1 4 4 4 4 4 3 1 4 4 3 4 4 4 3 4 1 3 1 1 1 4 3 3 1 3 4 1 4 4
##  [137] 3 3 4 3 4 3 4 4 3 4 3 3 3 4 4 4 4 3 4 4 3 4 3 4 4 4 4 3 4 4 4 3 4 3
##  [171] 4 4 3 4 4 3 4 3 3 4 3 4 3 4 3 4 4 4 3 4 2 3 3 3 4 3 3 2 2 3 3 3 3 3
##  [205] 3 3 3 3 3 2 2 2 2 3 3 2 2 2 2 2 3 3 3 3 2 3 3 3 3 3 2 2 3 3 3 3 3 3
##  [239] 2 3 2 2 3 3 3 3 2 1 3 1 1 3 1 4 1 2 1 4 3 1 3 4 3 1 3 2 4 2 4 3 4 4
##  [273] 4 2 3 3 3 2 3 3 3 3 1 2 3 3 3 4 4 3 3 3 3 3 3 3 3 3 3 2 3 3 3 3 3 3
##  [307] 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 4 4 4 4
##  [341] 4 4 1 1 1 1 2 4 1 3 4 4 3 1 4 3 4 3 1 2 4 4 1 2 4 2 4 4 3 1 2 2 3 3
##  [375] 2 2 3 4 3 3 4 3 4 3 4 3 4 2 4 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
##  [409] 4 4 4 4 4 4 4 4 4 4 4 4 4 4 1 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
##  [443] 4 4 4 4 4 4 4 4 4 4 4 4 4 1 4 4 4 1 4 4 4 4 1 4 1 4 1 4 4 4 4 4 4 4
##  [477] 4 4 4 4 4 1 4 4 4 4 4 4 4 4 4 1 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 4 3 1
##  [511] 4 4 4 4 4 4 4 4 4 4 1 4 4 4 4 1 4 4 3 4 3 4 3 4 3 4 3 4 4 4 1 4 1 1
##  [545] 4 1 4 3 4 3 4 3 4 3 4 1 4 3 4 3 4 3 4 3 1 1 1 1 4 3 3 4 3 4 3 4 3 1
##  [579] 1 1 3 4 3 4 1 4 3 4 2 1 4 1 3 1 1 3 3 2 3 3 3 3 3 3 3 2 3 2 3 2 3 3
##  [613] 3 3 3 3 2 3 2 3 3 1 2 3 2 3 2 3 2 3 2 2 2 1 3 3 2 3 2 3 2 2 4 3 3 4
##  [647] 3 3 4 4 3 3 2 3 4 3 3 4 3 4 3 4 1 4 3 4 3 4 3 4 3 4 3 4 3 4 3 4 2 4
##  [681] 3 4 2 4 3 4 4 3 4 2 4 2 4 3 4 3 4 1 4 3 4 3 4 3 4 2 4 3 4 3 4 3 4 3
##  [715] 4 3 4 3 4 3 1 1 4 3 3 4 3 3 3 1 4 4 1 4 1 4 4 1 4 3 4 3 3 3 4 3 1 3
##  [749] 1 3 1 3 4 4 4 4 4 3 4 4 4 3 1 3 1 3 3 3 1 3 3 1 3 4 3 1 3 3 4 3 4 3
##  [783] 4 4 3 4 3 4 3 1 3 4 3 4 1 3 1 4 1 4 1 4 1 4 4 4 4 4 4 4 1 4 4 4 4 4
##  [817] 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 1 4 1 4 4 4 4 4 4 4 4 1 4 4 1 2
##  [851] 3 3 3 2 3 1 4 3 4 4 4 4 4 3 3 3 3 3 3 3 4 3 2 3 3 3 3 3 3 3 3 3 3 2
##  [885] 2 2 3 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
##  [919] 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
##  [953] 3 3 3 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 3 2
##  [987] 2 2 3 2 3 3 3 3 3 3 3 3 3 2 2 2 2 3 3 2 3 2 2 3 2 2 2 3 2 2 2 2 2 2
## [1021] 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 2 2 2 3 3 3 3 2 2 2 3 3 3 3
## [1055] 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 2 3 3 3 3 3 3 3 3 3 3 3 3 2 3 3 2 1
## [1089] 4 3 2 1 4 1 1 4 1 1 1 4 4 1 1 1 1 4 4 4 1 1 1 1 4 1 1 1 1 4 4 1 1 4
## [1123] 4 4 1 4 1 4 1 4 4 1 4 4 4 4 1 4 4 4 1 1 4 1 1 1 4 4 4 1 4 1 4 1 4 4
## [1157] 1 1 4 1 4 4 1 4 4 1 3 4 4 4 1 1 1 1 4 1 4 1 4 4 4 1 4 1 1 1 3 4 1 1
## [1191] 1 4 1 1 1 4 1 1 1 1 1 1 4 1 1 1 1 1 4 1 4 4 1 1 1 4 4 1 4 1 1 1 1 1
## [1225] 1 4 4 4 4 4 1 4 1 1 1 1 1 1 4 1 4 4 1 1 1 1 1 1 1 4 4 1 4 4 4 4 1 3
## [1259] 1 4 4 3 4 3 1 3 1 3 1 1 3 2 4 3 4 2 4 3 1 2 4 3 1 3 1 3 4 4 1 3 1 1
## [1293] 1 1 3 4 4 4 1 3 4 4 4 4 1 3 1 4 1 1 4 1 4 1 4 4 1 4 1 4 4 4 4 4 4 4
## [1327] 4 1 4 4 4 4 1 4 1 1 4 1 4 1 4 1 4 4 4 4 4 4 4 1 4 4 4 4 4 4 4 1 4 4
## [1361] 4 4 4 4 4 4 4 2 4 4 3 3 1 1 1 1 1 3 3 3 4 3 3 3 3 3 3 1 3 3 3 4 3 3
## [1395] 3 4 3 1 3 3 2 4 3 3 3 4 2 4 4 2 4 3 4 3 4 4 2 4 4 2 2 4 4 4 1 4 4 4
## [1429] 1 4 1 1 4 3 4 3 1 3 4 3 4 3 1 3 4 3 4 3 4 3 3 3 4 2 4 3 4 3 4 3 4 3
## [1463] 4 3 4 3 4 3 3 3 3 3 3 1 3 1 3 3 4 3 3 4 3 4 4 3 4 3 4 3 4 3 4 3 4 3
## [1497] 3 4 4 4 3 3 3 4 3 1 3 3 3 3 4 3 4 4 3 4 2 3 4 4 3 4 3 4 3 3 3 3 3 1
## [1531] 3 3 4 3 3 3 3 3 1 3 3 1 3 3 4 3 3 3 4 1 4 4 4 4 4 2 4 4 3 4 3 4 2 4
## [1565] 2 4 2 4 4 4 2 4 3 4 2 4 3 4 2 4 3 4 3 4 2 4 4 3 4 4 4 2 4 4 2 4 4 1
## [1599] 4 2 4 2 4 3 2 2 4 4 4 4 3 4 4 3 4 4 4 3 4 3 4 1 4 3 4 4 4 1 3 3 3 4
## [1633] 3 1 3 1 3 4 4 4 4 4 3 1 3 2 4 3 3 1 1 1 3 1 1 3 2 3 3 3 3 3 1 3 1 3
## [1667] 1 3 1 2 4 3 4 2 4 3 4 3 4 3 1 1 3 3 4 3 4 3 1 3 4 4 3 1 4 4 3 1 4 3
## [1701] 3 3 3 1 3 4 3 3 4 3 1 4 3 4 3 3 4 3 3 4 3 3 3 4 3 3 4 3 3 3 3 3 3 3
## [1735] 3 3 3 3 3 3 3 3 3 3 2 3 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
## [1769] 3 3 4 3 4 1 3 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 3 3 3 3 3 3 3 3 3 3 3
## [1803] 3 3 3 3 3 3 3 2 1 4 4 3 1 4 2 1 4 4 4 4 4 3 3 1 4 2 4 2 4 4 4 4 4 4
## [1837] 4 4 4 4 4 4 1 4 4 3 4 3 3 3 4 4 4 4 3 3 3 3 3 3 3 3 3 3 2 3 3 2 3 2
## [1871] 2 2 2 2 2 2 3 3 3 3 3 3 3 2 3 3 1 2 2 2 3 3 2 3 2 1 3 3 2 2 1 2 2 1
## [1905] 3 3 3 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 3 2 3 2 2 2 2 2 2 2 2 2 3
## [1939] 2 2 3 3 3 2 3 3 3 2 3 3 2 3 3 3 3 3 3 3 3 3 3 2 2 2 3 3 2 2 2 3 2 3
## [1973] 3 3 3 3 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 3 3 3 3 3
## [2007] 3 3 3 3 3 3 3 3 3 2 2 3 3 2 2 3 2 3 2 3 3 3 3 3 3 3 2 2 2 2 2 3 2 2
## [2041] 2 2 2 2 2 2 2 3 2 2 2 3 2 3 2 2 3 2 2 2 2 2 3 2 2 2 2 3 3 3 2 2 2 2
## [2075] 2 2 2 3 3 2 2 2 2 2 2 2 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3
## [2109] 2 3 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 2 1
## 
## Within cluster sum of squares by cluster:
## [1]  59.07688  54.63678  73.23830 115.71393
##  (between_SS / total_SS =  94.1 %)
## 
## Available components:
## 
## [1] "cluster"      "centers"      "totss"        "withinss"    
## [5] "tot.withinss" "betweenss"    "size"         "iter"        
## [9] "ifault"
fviz_cluster(clusters, data=nyc.env)

#Conduct a gap_statistic analysis to determine optimal number of clusters
gap_stat<-clusGap(nyc.env, FUN=kmeans, nstart=25, K.max=9, B=50)
print(gap_stat, method="firstmax")
## Clustering Gap statistic ["clusGap"] from call:
## clusGap(x = nyc.env, FUNcluster = kmeans, K.max = 9, B = 50,     nstart = 25)
## B=50 simulated reference sets, k = 1..9; spaceH0="scaledPCA"
##  --> Number of clusters (method 'firstmax'): 8
##           logW   E.logW       gap      SE.sim
##  [1,] 6.818152 7.377546 0.5593943 0.005985380
##  [2,] 5.854314 7.149895 1.2955808 0.005929672
##  [3,] 5.647194 6.991090 1.3438961 0.005486990
##  [4,] 5.502733 6.854749 1.3520154 0.004973083
##  [5,] 5.413318 6.794889 1.3815714 0.004773167
##  [6,] 5.356176 6.742573 1.3863974 0.004890486
##  [7,] 5.304725 6.698461 1.3937361 0.004408503
##  [8,] 5.262788 6.658238 1.3954499 0.003981742
##  [9,] 5.248153 6.621767 1.3736139 0.004295583
clusters.1<-kmeans(nyc.env, 8, nstart=25)
str(clusters.1)
## List of 9
##  $ cluster     : int [1:2140] 4 4 4 4 5 1 5 4 5 8 ...
##  $ centers     : num [1:8, 1:4] 0.487 0.245 0.276 0.42 0.319 ...
##   ..- attr(*, "dimnames")=List of 2
##   .. ..$ : chr [1:8] "1" "2" "3" "4" ...
##   .. ..$ : chr [1:4] "Pct.Low.Inc" "Diesel.Conc" "NPL.Site.Count..wt.by.dist" "Pct.pre1960.Housing"
##  $ totss       : num 5120
##  $ withinss    : num [1:8] 27.6 24.6 23.5 33.2 38.2 ...
##  $ tot.withinss: num 188
##  $ betweenss   : num 4933
##  $ size        : int [1:8] 473 276 420 280 234 2 136 319
##  $ iter        : int 3
##  $ ifault      : int 0
##  - attr(*, "class")= chr "kmeans"
clusters.1
## K-means clustering with 8 clusters of sizes 473, 276, 420, 280, 234, 2, 136, 319
## 
## Cluster means:
##   Pct.Low.Inc Diesel.Conc NPL.Site.Count..wt.by.dist Pct.pre1960.Housing
## 1   0.4872041    4.782577                  0.1515335           0.8077177
## 2   0.2448580    5.110912                  0.2557669           0.8033957
## 3   0.2759884    4.408634                  0.1095319           0.7970079
## 4   0.4198341    1.620486                  0.1240337           0.4135573
## 5   0.3190396    2.326111                  0.1906905           0.3905817
## 6   0.1281830    5.238331                  3.8956107           0.8469171
## 7   0.2115952    1.011876                  0.1200858           0.2921808
## 8   0.4667308    5.426907                  0.1408217           0.7866891
## 
## Clustering vector:
##    [1] 4 4 4 4 5 1 5 4 5 8 5 4 1 5 1 5 4 5 1 5 4 5 1 4 1 1 1 8 1 5 1 1 8 1
##   [35] 5 1 8 4 8 8 4 5 1 5 1 1 4 5 1 4 1 4 1 4 1 1 4 1 1 1 4 1 1 4 4 4 7 4
##   [69] 4 4 4 4 4 4 4 4 4 1 1 7 5 8 4 8 8 8 1 1 1 4 1 4 4 4 7 1 1 1 5 1 5 5
##  [103] 5 8 8 8 5 8 8 8 8 8 1 5 8 8 1 8 8 8 1 8 4 1 5 5 4 8 1 1 5 1 8 5 8 8
##  [137] 1 1 8 1 8 1 8 8 1 8 1 1 1 8 8 8 8 1 8 8 1 8 1 8 8 8 8 1 1 8 8 1 8 1
##  [171] 8 8 1 8 8 1 8 1 1 8 1 8 1 8 1 8 2 8 1 1 4 1 1 1 1 1 1 4 4 1 3 3 1 1
##  [205] 3 1 3 1 3 4 4 4 4 3 1 4 4 4 4 4 3 3 3 3 7 3 1 3 1 1 4 4 1 3 1 3 1 3
##  [239] 4 3 4 7 3 3 3 3 4 4 3 4 4 3 4 1 4 4 4 1 3 4 3 1 1 4 3 4 8 4 8 3 8 1
##  [273] 1 4 1 3 1 4 1 3 1 3 5 4 1 3 1 1 1 1 1 1 3 1 1 3 1 1 3 4 3 1 3 1 3 3
##  [307] 3 3 3 3 3 1 3 3 1 1 3 1 3 1 3 3 3 3 3 3 3 3 3 1 3 4 4 4 7 7 8 1 2 2
##  [341] 2 2 5 5 5 5 4 1 5 1 8 8 3 5 2 3 2 3 5 4 2 2 5 4 2 4 2 2 3 5 4 4 3 3
##  [375] 4 4 1 2 1 1 2 1 2 3 2 1 2 4 2 1 1 6 1 6 1 1 1 1 2 1 1 1 1 1 1 1 2 1
##  [409] 1 1 1 1 1 1 1 2 1 2 1 2 1 1 5 2 2 2 1 2 2 2 2 2 8 2 8 2 8 2 8 2 2 2
##  [443] 8 2 8 2 8 2 2 2 2 2 2 2 2 5 2 8 2 5 2 8 2 8 5 2 5 8 5 8 2 8 2 8 8 8
##  [477] 2 8 8 2 1 5 1 2 8 2 8 2 8 2 8 5 8 2 8 2 2 2 8 1 1 1 2 1 1 1 1 1 1 4
##  [511] 1 1 1 1 2 1 2 1 2 1 5 1 1 1 2 5 1 1 1 8 1 8 1 8 1 8 1 8 1 8 4 8 4 4
##  [545] 8 4 8 1 8 1 8 1 8 1 8 4 8 1 8 1 8 1 8 1 5 4 5 4 8 1 1 2 1 8 1 8 1 5
##  [579] 4 5 1 8 1 8 4 8 1 2 4 5 1 5 1 5 4 1 1 4 1 1 1 1 1 1 1 4 1 4 1 4 1 1
##  [613] 1 3 1 1 4 1 4 1 1 4 4 1 4 1 4 1 4 1 4 4 4 4 3 1 4 1 4 1 4 4 1 3 1 2
##  [647] 1 1 2 1 1 1 4 1 8 1 1 2 1 2 1 2 4 8 1 8 1 8 1 2 1 8 1 8 1 8 1 8 4 8
##  [681] 1 2 4 8 1 2 8 1 8 4 8 4 8 1 8 1 8 4 8 1 8 1 8 1 8 4 8 1 8 1 8 1 8 1
##  [715] 8 1 8 1 8 1 5 5 1 2 1 2 1 1 1 4 1 1 5 1 4 1 1 5 1 1 2 1 1 1 1 1 4 1
##  [749] 4 1 4 1 2 1 2 1 2 2 2 2 2 2 4 1 4 1 1 1 4 1 1 5 1 2 1 5 1 1 2 1 2 1
##  [783] 8 8 1 8 1 8 1 5 1 8 1 8 5 1 5 8 5 8 5 8 5 8 8 8 8 8 2 8 5 8 8 8 8 8
##  [817] 8 8 2 8 2 2 2 2 2 1 8 2 8 2 2 8 8 8 5 2 5 8 2 2 2 2 2 2 2 4 2 1 4 4
##  [851] 1 1 1 4 1 4 1 1 2 2 2 2 2 1 1 1 3 3 3 3 2 1 4 3 3 3 3 3 3 3 3 3 3 4
##  [885] 4 4 3 4 4 4 7 4 3 3 3 3 3 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1
##  [919] 1 1 3 1 3 1 1 1 1 1 1 1 1 1 1 1 1 1 4 1 1 1 1 1 1 1 1 1 1 3 3 1 3 3
##  [953] 3 1 1 4 1 1 1 1 1 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4 4 4 4 4 4 1 4
##  [987] 4 4 3 4 3 3 3 3 3 1 3 1 3 4 4 4 4 3 3 4 3 4 4 1 4 4 4 3 4 4 4 4 4 4
## [1021] 4 4 3 4 4 4 4 4 4 4 7 7 4 4 7 4 3 3 1 1 4 4 4 3 3 1 3 4 4 4 3 1 1 1
## [1055] 4 4 4 4 4 4 4 4 1 1 1 1 3 1 1 3 4 3 1 1 3 3 1 1 3 1 1 3 3 4 1 3 7 5
## [1089] 2 1 4 5 8 5 5 8 5 5 5 8 2 5 5 5 5 8 8 8 5 5 5 5 8 5 5 5 5 8 8 5 5 8
## [1123] 2 8 5 8 5 8 5 8 8 5 8 8 2 2 5 2 8 8 5 5 2 5 5 5 2 8 8 5 8 5 2 5 2 8
## [1157] 5 5 8 5 2 8 5 8 8 5 3 8 8 8 5 5 5 5 8 5 8 5 8 8 8 5 8 5 5 5 3 8 5 5
## [1191] 5 8 5 5 5 8 5 5 5 5 5 5 2 5 5 5 5 5 8 5 8 2 5 5 5 8 2 5 2 5 5 5 5 5
## [1225] 5 8 2 8 2 2 4 2 5 5 5 5 5 5 2 5 2 2 5 5 5 5 5 5 4 2 2 5 2 2 2 2 5 2
## [1259] 4 2 1 2 1 2 4 2 4 2 4 5 2 4 2 2 1 4 8 2 5 4 1 1 5 1 5 2 8 1 5 1 5 5
## [1293] 5 5 1 2 2 2 5 3 8 2 8 8 5 1 5 8 5 5 2 5 2 4 8 8 5 2 4 8 2 8 8 8 2 8
## [1327] 8 5 2 8 8 8 5 2 5 5 8 5 2 5 8 5 8 8 8 8 8 2 2 5 8 8 8 8 8 8 8 5 8 8
## [1361] 8 8 8 8 8 8 8 4 8 2 1 1 5 5 5 5 5 3 3 3 8 3 3 3 3 1 3 5 3 3 3 8 3 3
## [1395] 3 8 3 5 3 3 4 8 3 3 3 8 4 2 8 7 8 3 2 3 8 8 4 2 2 7 7 2 2 2 5 2 2 2
## [1429] 5 2 5 5 8 3 1 3 5 3 2 3 2 3 5 3 2 3 2 3 2 3 3 3 2 4 2 3 1 3 2 3 2 3
## [1463] 2 3 2 3 2 3 3 3 3 3 3 5 1 5 1 3 2 3 3 2 3 2 2 3 2 3 2 3 2 3 2 3 8 3
## [1497] 3 8 8 8 3 3 3 8 3 5 3 3 3 3 8 3 8 8 3 8 4 3 8 8 3 8 3 8 3 3 3 3 3 5
## [1531] 3 3 2 3 3 1 3 1 5 1 3 5 1 3 8 3 1 1 8 5 8 8 8 8 8 4 8 8 3 8 3 8 4 8
## [1565] 4 8 4 8 8 8 4 8 3 8 4 2 3 2 4 2 3 2 3 2 4 2 2 3 2 8 8 4 2 2 7 2 2 5
## [1599] 2 7 2 7 2 3 7 7 2 2 2 2 3 2 2 3 8 2 2 3 2 3 2 5 8 3 8 8 8 5 3 3 3 8
## [1633] 3 5 3 5 3 8 8 8 8 8 3 5 3 4 8 3 3 5 5 5 3 5 5 3 4 3 1 3 3 3 5 3 5 3
## [1667] 5 3 5 4 8 3 8 4 8 3 8 3 8 3 5 5 3 3 2 3 8 3 5 3 8 8 3 5 8 2 3 5 8 3
## [1701] 3 3 3 5 3 2 3 3 2 3 5 8 3 8 3 3 2 3 3 2 3 3 3 2 3 3 1 3 3 1 3 3 1 1
## [1735] 3 1 1 3 1 3 1 3 1 3 4 3 4 3 3 1 3 1 3 1 1 3 1 3 1 1 1 1 3 1 3 1 3 2
## [1769] 3 3 2 3 2 4 3 5 3 3 3 1 3 3 3 3 3 1 3 1 3 1 7 3 3 3 3 3 3 3 3 1 2 3
## [1803] 3 3 3 2 3 3 3 7 5 2 2 3 5 2 7 5 2 2 2 2 2 3 3 5 2 7 2 7 2 2 2 2 2 2
## [1837] 2 2 2 2 2 2 5 2 2 2 2 2 2 2 2 2 2 2 1 1 2 1 1 1 1 1 1 1 4 1 1 4 3 4
## [1871] 4 4 4 4 4 4 1 1 3 3 1 3 3 4 3 3 4 4 4 4 1 3 4 3 4 4 1 3 4 7 4 7 7 5
## [1905] 3 1 3 4 7 7 7 4 7 7 7 4 1 7 7 7 7 7 4 4 7 3 7 3 7 7 7 7 7 7 7 7 7 3
## [1939] 7 7 3 3 3 4 3 3 3 7 3 3 7 3 3 3 3 3 3 3 3 3 3 4 4 4 3 3 4 4 4 3 4 3
## [1973] 3 3 3 3 4 3 3 3 1 1 3 3 1 1 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 3 3 3 3 3
## [2007] 3 3 3 3 3 3 3 3 3 4 7 3 3 7 7 3 7 3 7 3 3 3 3 3 3 3 4 7 4 7 4 3 4 7
## [2041] 7 7 4 4 7 7 7 3 7 7 7 3 7 3 7 7 3 4 4 7 7 4 3 7 7 7 7 3 3 3 7 7 7 7
## [2075] 7 7 4 3 3 7 7 7 7 7 7 7 3 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 3
## [2109] 7 3 7 7 7 7 3 4 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 4 7 7 7 7 3 4 5
## 
## Within cluster sum of squares by cluster:
## [1] 27.645832 24.618364 23.497208 33.164919 38.231521  4.285446 11.894939
## [8] 24.261436
##  (between_SS / total_SS =  96.3 %)
## 
## Available components:
## 
## [1] "cluster"      "centers"      "totss"        "withinss"    
## [5] "tot.withinss" "betweenss"    "size"         "iter"        
## [9] "ifault"
fviz_cluster(clusters.1, data=nyc.env)

The next thing that I did was Hierarichical clustering setting the distance Euclidian methods since the data was all continuous and numeric. A dissimilarity matrix was used to get the distance and looked at the difference clustering methods: Complete, Single, and average A gap statistic was also used to get the optimal number of clusters.

set.seed(100)

# Create Dissimilarity matrix
diss.matrix <- dist(nyc.env, method = "euclidean")

gap_stat <- clusGap(nyc.env, FUN = hcut, nstart = 25, K.max = 10, B = 50, mc_metric = "euclidean")
fviz_gap_stat(gap_stat)

gap_stat1 <- clusGap(nyc.env, FUN = hcut, nstart = 25, K.max = 10, B = 50, mc_metric = "euclidean", method = "single")
fviz_gap_stat(gap_stat1)

# Create Dissimilarity matrix
diss.matrix <- dist(nyc.env, method = "euclidean")

# Hierarchical clustering using Single Linkage
hc1 <- hclust(diss.matrix, method = "single" )

groups.1<-cutree(hc1,4)
aggregate(nyc.env,list(groups.1),mean)
##   Group.1 Pct.Low.Inc Diesel.Conc NPL.Site.Count..wt.by.dist
## 1       1  0.33997812    1.747171                  0.1472041
## 2       2  0.37824636    4.876062                  0.1567154
## 3       3  0.09781744    5.234196                  2.4328626
## 4       4  0.15854864    5.242467                  5.3583589
##   Pct.pre1960.Housing
## 1           0.3798904
## 2           0.7993850
## 3           0.8933042
## 4           0.8005300
# Plot the obtained dendrogram using Single
plot(hc1, cex = 0.6, hang = -1)

gap_stat2 <- clusGap(nyc.env, FUN = hcut, nstart = 25, K.max = 10, B = 50, mc_metric = "euclidean", method = "complete")
fviz_gap_stat(gap_stat2)

# Hierarchical clustering using Complete Linkage
hc2 <- hclust(diss.matrix, method = "complete" )

groups.2 <-cutree(hc2,4)
aggregate(nyc.env,list(groups.2),mean)
##   Group.1 Pct.Low.Inc Diesel.Conc NPL.Site.Count..wt.by.dist
## 1       1   0.3422037    1.728378                  0.1474176
## 2       2   0.3782615    4.876946                  0.1582655
## 3       3   0.1585486    5.242467                  5.3583589
## 4       4   0.1708487    3.169033                  0.1314285
##   Pct.pre1960.Housing
## 1           0.3801287
## 2           0.7999302
## 3           0.8005300
## 4           0.3348217
# Plot the obtained dendrogram using Complete
plot(hc2, cex = 0.6, hang = -1)

gap_stat3 <- clusGap(nyc.env, FUN = hcut, nstart = 25, K.max = 10, B = 50, mc_metric = "euclidean", method = "average")
fviz_gap_stat(gap_stat3)

# Hierarchical clustering using Average Linkage
hc3 <- hclust(diss.matrix, method = "average" )

groups.3<-cutree(hc3,4)
aggregate(nyc.env,list(groups.3),mean)
##   Group.1 Pct.Low.Inc Diesel.Conc NPL.Site.Count..wt.by.dist
## 1       1   0.3399781    1.747171                  0.1472041
## 2       2   0.3784319    4.875831                  0.1550308
## 3       3   0.1928457    5.109687                  1.7498787
## 4       4   0.1585486    5.242467                  5.3583589
##   Pct.pre1960.Housing
## 1           0.3798904
## 2           0.7993745
## 3           0.8358926
## 4           0.8005300
# Plot the obtained dendrogram using Average
plot(hc3, cex = 0.6, hang = -1)

Question 2: Describe the outputs of the analysis in terms of their composition of the input features.

The optimal number of cluster based on the k means clustering plot are eight cluster, this is inconsistent with the optimal number of cluster based on hierarchical clustering. For this type of clustering, the gap statistic stated that five was the optimal number of cluster to have. Five clusters was consistent acroos the different types of linkage methods, single, complete and average. Although the cluster dendrograms all look different the results from all of them are the same.

The means for perecnt of low income and percentage of housing stock built before 1960 are similar across all the linkaage methods. The means observed were similar to each other, there was a little more varaition in the means but not a huge varaition for air concentration of diesal extract and proximity to hazardous waste sites.

Question 3: List a subsequent research question that could be addressed using the output of this analysis. The output can be used as an exposure, outcome or confounding variable. State what type of research question it is: descriptive, explanatory or predictive.

A subsequential research question that can be addressed using the outbut of this analysis is a explanatory question which is if a proportion of housing stocking built before 1960 as a proxy for potential lead exposure is associated with increase ambient air concentrations of diesel exhaust, while controling for proportion of neighborhood with household income lwoer than the city median.

Part 2: Choose your own supervised adventure

B) Researchers are interested in uncovering nutritional factors that influence the development of diabetes. To generate hypotheses to guide future research, they ask you to utilize feature selection methods to identify the factors that are most relevant to fasting blood glucose. (The dataset you are given is glucose.csv.) You should try two different algorithms for feature selection and comment if they identify the same set of variables as being important for serum glucose.

The first thing that will be done is import and clean the data.

glucose.raw<-read.csv("glucose.csv", header=TRUE)

glucose =
  glucose.raw %>%  
  drop_na() %>% 
  janitor::clean_names()

Next I will partition the data so that there is a 70/30 split and set the seed to 100.

set.seed(100)
training.data.glucose<-glucose$lbxsgl %>% createDataPartition(p=0.7, list=F)
train.data.glucose<-glucose[training.data.glucose, ]
test.data.glucose<-glucose[-training.data.glucose, ]

#Store outcome 
glucose.train<-train.data.glucose$lbxsgl
glucose.test<-test.data.glucose$lbxsgl

# Store the outcome in one train and test and the predictors in another 
#model matrix- will create indicator variables for categorical varaibles, it does not do anything to the continuous variables
x.train<-model.matrix(lbxsgl~., train.data.glucose)[,-1]
x.test<-model.matrix(lbxsgl~., test.data.glucose)[,-1]

The next thing that I did was create different regression trees to find the one with the best fit. I tried different CPs to find the best one. I double checked this using caret to see if I got similar results.

#Regression Tree 1-default values of rpart
tree.glucose<-rpart(lbxsgl ~ ., data=train.data.glucose, method="anova")
plot(tree.glucose, uniform=TRUE)
text(tree.glucose, use.n=TRUE, all=TRUE, cex=0.8)

printcp(tree.glucose)
## 
## Regression tree:
## rpart(formula = lbxsgl ~ ., data = train.data.glucose, method = "anova")
## 
## Variables actually used in tree construction:
## [1] bmxwaist lbxsse   ridageyr x       
## 
## Root node error: 921862/2416 = 381.57
## 
## n= 2416 
## 
##         CP nsplit rel error  xerror    xstd
## 1 0.054087      0   1.00000 1.00075 0.13340
## 2 0.021376      1   0.94591 0.95737 0.13163
## 3 0.013929      4   0.88179 0.95906 0.13120
## 4 0.010000      5   0.86786 0.94416 0.12774
plotcp(tree.glucose)

print(tree.glucose)
## n= 2416 
## 
## node), split, n, deviance, yval
##       * denotes terminal node
## 
##  1) root 2416 921862.30  93.78891  
##    2) ridageyr< 41.5 1387 340232.70  89.87599  
##      4) bmxwaist< 100.75 1053 133523.20  87.83761 *
##      5) bmxwaist>=100.75 334 188540.50  96.30240  
##       10) lbxsse< 147.45 298  69364.45  94.48658 *
##       11) lbxsse>=147.45 36 110060.00 111.33330  
##         22) x>=1368.5 29  26209.79  96.72414 *
##         23) x< 1368.5 7  52018.86 171.85710 *
##    3) ridageyr>=41.5 1029 531768.90  99.06317  
##      6) bmxwaist< 92.85 366  80156.11  94.30874 *
##      7) bmxwaist>=92.85 663 438772.40 101.68780 *
rpart.plot(tree.glucose)

#Regression Tree 2- varying values in rpart.control, specifically going to cp=0.001, to find minimum cv-error
tree.glucose.1<-rpart(lbxsgl ~ ., data=train.data.glucose, method="anova", control=rpart.control(cp=0.001))
plotcp(tree.glucose.1)

rpart.plot(tree.glucose.1)

printcp(tree.glucose.1)
## 
## Regression tree:
## rpart(formula = lbxsgl ~ ., data = train.data.glucose, method = "anova", 
##     control = rpart.control(cp = 0.001))
## 
## Variables actually used in tree construction:
##  [1] bmxbmi   bmxht    bmxwaist bmxwt    lbxlyc   lbxmg    lbxscu  
##  [8] lbxsse   lbxszn   riagendr ridageyr ridreth1 x       
## 
## Root node error: 921862/2416 = 381.57
## 
## n= 2416 
## 
##           CP nsplit rel error  xerror    xstd
## 1  0.0540870      0   1.00000 1.00073 0.13329
## 2  0.0213757      1   0.94591 0.95896 0.13027
## 3  0.0139288      4   0.88179 0.97858 0.12850
## 4  0.0072132      5   0.86786 1.01453 0.12867
## 5  0.0071002     10   0.83179 1.05593 0.12974
## 6  0.0056439     17   0.78209 1.05906 0.12998
## 7  0.0036642     18   0.77645 1.05516 0.12792
## 8  0.0035573     19   0.77278 1.07097 0.12854
## 9  0.0032644     20   0.76922 1.07372 0.12855
## 10 0.0029512     21   0.76596 1.07968 0.12884
## 11 0.0028427     22   0.76301 1.08177 0.12897
## 12 0.0028339     26   0.75132 1.08180 0.12897
## 13 0.0027661     27   0.74849 1.08156 0.12903
## 14 0.0025352     30   0.74019 1.09059 0.12938
## 15 0.0023634     31   0.73766 1.10767 0.13036
## 16 0.0023415     32   0.73529 1.10649 0.13035
## 17 0.0023339     33   0.73295 1.10649 0.13035
## 18 0.0022611     34   0.73062 1.10703 0.13035
## 19 0.0020648     35   0.72836 1.10970 0.13040
## 20 0.0019484     38   0.72216 1.11060 0.12979
## 21 0.0019062     39   0.72021 1.10780 0.12800
## 22 0.0018994     40   0.71831 1.10754 0.12800
## 23 0.0017666     41   0.71641 1.11037 0.12825
## 24 0.0017172     45   0.70906 1.11497 0.12840
## 25 0.0016434     48   0.70391 1.11504 0.12840
## 26 0.0015625     49   0.70227 1.11615 0.12851
## 27 0.0015559     50   0.70070 1.11638 0.12833
## 28 0.0015129     54   0.69448 1.11745 0.12842
## 29 0.0014334     55   0.69297 1.11802 0.12840
## 30 0.0013583     56   0.69153 1.12057 0.12839
## 31 0.0013425     59   0.68746 1.12503 0.12813
## 32 0.0012050     60   0.68612 1.12660 0.12775
## 33 0.0011754     61   0.68491 1.13106 0.12788
## 34 0.0011659     62   0.68374 1.13120 0.12788
## 35 0.0011652     63   0.68257 1.13120 0.12788
## 36 0.0011439     65   0.68024 1.13194 0.12787
## 37 0.0011432     66   0.67910 1.13350 0.12788
## 38 0.0011007     68   0.67681 1.13255 0.12766
## 39 0.0010868     69   0.67571 1.13272 0.12766
## 40 0.0010857     72   0.67245 1.13256 0.12766
## 41 0.0010821     73   0.67136 1.13284 0.12770
## 42 0.0010711     74   0.67028 1.13460 0.12770
## 43 0.0010703     78   0.66600 1.13684 0.12770
## 44 0.0010489     80   0.66386 1.14031 0.12772
## 45 0.0010257     81   0.66281 1.14350 0.12773
## 46 0.0010253     82   0.66178 1.14436 0.12779
## 47 0.0010000     83   0.66076 1.14497 0.12778
selected.cp<-tree.glucose.1$cptable[which.min(tree.glucose.1$cptable[,"xerror"]), "CP"]

tree.glucose.pruned<-prune(tree.glucose.1, cp=selected.cp)
rpart.plot(tree.glucose.pruned)

#Regression Tree 3-manually setting cp to 1SD greater error than minimum value
tree.glucose.2<-rpart(lbxsgl ~ ., data=train.data.glucose, method="anova", control=rpart.control(cp=0.006665509))
rpart.plot(tree.glucose.2)

print(tree.glucose.2)
## n= 2416 
## 
## node), split, n, deviance, yval
##       * denotes terminal node
## 
##    1) root 2416 921862.300  93.78891  
##      2) ridageyr< 41.5 1387 340232.700  89.87599  
##        4) bmxwaist< 100.75 1053 133523.200  87.83761 *
##        5) bmxwaist>=100.75 334 188540.500  96.30240  
##         10) lbxsse< 147.45 298  69364.450  94.48658 *
##         11) lbxsse>=147.45 36 110060.000 111.33330  
##           22) x>=1368.5 29  26209.790  96.72414 *
##           23) x< 1368.5 7  52018.860 171.85710 *
##      3) ridageyr>=41.5 1029 531768.900  99.06317  
##        6) bmxwaist< 92.85 366  80156.110  94.30874 *
##        7) bmxwaist>=92.85 663 438772.400 101.68780  
##         14) x< 4741 656 382603.000 101.37040  
##           28) ridreth1>=1.5 579 275652.800 100.22450  
##             56) lbxszn>=55.25 570 239681.100  99.82281  
##              112) bmxwaist< 126.35 541 208123.400  99.10166  
##                224) ridageyr< 69.5 403  95976.460  97.15385 *
##                225) ridageyr>=69.5 138 106152.900 104.78990  
##                  450) lbxmg>=2.043592 54   9864.833  97.05556 *
##                  451) lbxmg< 2.043592 84  90981.240 109.76190  
##                    902) lbxlyc< 0.3839563 28   3906.107  98.67857 *
##                    903) lbxlyc>=0.3839563 56  81915.840 115.30360  
##                     1806) x< 1740 20   4594.950 101.45000 *
##                     1807) x>=1740 36  71350.000 123.00000  
##                       3614) lbxsse>=128.95 25  12245.840 110.92000 *
##                       3615) lbxsse< 128.95 11  47164.730 150.45450 *
##              113) bmxwaist>=126.35 29  26027.790 113.27590 *
##             57) lbxszn< 55.25 9  30054.000 125.66670 *
##           29) ridreth1< 1.5 77 100473.000 109.98700  
##             58) riagendr>=1.5 32   9049.500  99.87500 *
##             59) riagendr< 1.5 45  85824.580 117.17780  
##              118) lbxscu< 98.4 14   4297.714  98.14286 *
##              119) lbxscu>=98.4 31  74163.420 125.77420  
##                238) ridageyr>=53.5 20   8725.200 114.20000 *
##                239) ridageyr< 53.5 11  57887.640 146.81820 *
##         15) x>=4741 7  49911.710 131.42860 *
#Fit model to test data and calculate R-squared and MSE
pred.intest<-predict(tree.glucose.2, newdata=test.data.glucose)
r.square.test<-cor(test.data.glucose$lbxsgl, pred.intest)^2
r.square.test
## [1] 0.04498644
pred.mse<-mean((pred.intest-test.data.glucose$lbxsgl)^2)

#Plot distribution of error (not squared error)
test<-data.frame(pred.intest-test.data.glucose$lbxsgl)
histogram(test$pred.intest...test.data.glucose.lbxsgl)

## Good to see what is happening and making sure you know where you have error, want to see if its normal and if the data is skewwed 
tree.glucose.2$variable.importance
##  ridageyr  bmxwaist         x      seqn    bmxbmi    lbxsse     bmxwt 
## 64457.323 47582.261 46811.074 46756.676 34075.066 28304.483 25320.586 
##    lbxscu    lbxlyc    lbxszn     bmxht  ridreth1  riagendr     lbxmg 
## 18436.316 14785.501 10060.125  9248.844  8020.843  5598.909  5477.824

In this section I double checked using caret to see if I got similar results as my regression trees above.

set.seed(100)
train.control<-trainControl(method="cv", number=10)
tree.glucose.3<-train(lbxsgl~ . , data=train.data.glucose, method="rpart",trControl=train.control)
tree.glucose.3$bestTune
##           cp
## 1 0.01392877
rpart.plot(tree.glucose.3$finalModel)

pred.intest.temp<-predict(tree.glucose.3, newdata=test.data.glucose)
r.square.test.temp<-cor(test.data.glucose$lbxsgl, pred.intest.temp)^2
r.square.test.temp
## [1] 0.04955636
#Specify tuneGrid so caret explores wider variety of cp-values
grid<-expand.grid(cp=seq(0.001,0.1, by=0.001))
tree.glucose.4<-train(lbxsgl ~ ., data=train.data.glucose, method="rpart", trControl=train.control, tuneGrid=grid)
rpart.plot(tree.glucose.4$finalModel)

tree.glucose.4$bestTune
##       cp
## 23 0.023
grid<-expand.grid(.cp=0.0067)
tree.glucose.5<-train(lbxsgl ~ ., data=train.data.glucose, method="rpart", trControl=train.control, tuneGrid=grid)
rpart.plot(tree.glucose.5$finalModel)

pred.intest.2<-predict(tree.glucose.5, newdata=test.data.glucose)

r.square.test.2<-cor(test.data.glucose$lbxsgl, pred.intest.2)^2
r.square.test.2
## [1] 0.04498644
pred.mse.2<-mean((pred.intest.2-test.data.glucose$lbxsgl)^2)

varImp(tree.glucose.5)
## rpart variable importance
## 
##          Overall
## seqn     100.000
## x        100.000
## lbxscu    55.203
## lbxsse    44.578
## ridageyr  42.895
## bmxht     34.640
## lbxlyc    33.463
## bmxwaist  28.422
## bmxbmi    23.774
## bmxwt     16.232
## lbxszn     4.004
## ridreth1   3.926
## lbxmg      1.580
## riagendr   0.000

I then did a linear regression model to see how that model was fitted.

model.glucuse <- lm(lbxsgl ~ ., data=train.data.glucose)
summary(model.glucuse)
## 
## Call:
## lm(formula = lbxsgl ~ ., data = train.data.glucose)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -43.772  -9.014  -2.066   4.697 233.161 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 49.9356330 59.1706995   0.844 0.398795    
## x           -0.0068140  0.0041561  -1.640 0.101231    
## seqn         0.0016057  0.0008884   1.807 0.070841 .  
## lbxscu      -0.0171197  0.0138677  -1.234 0.217138    
## lbxsse       0.0711291  0.0218334   3.258 0.001138 ** 
## lbxmg       -0.2764060  0.2709803  -1.020 0.307820    
## lbxlyc       1.3169067  1.2999037   1.013 0.311124    
## lbxszn      -0.0515764  0.0249400  -2.068 0.038746 *  
## bmxwt        0.3232066  0.1577145   2.049 0.040540 *  
## bmxht       -0.4945555  0.1474998  -3.353 0.000812 ***
## bmxbmi      -0.9720705  0.4442199  -2.188 0.028746 *  
## bmxwaist     0.2593932  0.0747389   3.471 0.000528 ***
## riagendr    -4.2378956  1.0510519  -4.032  5.7e-05 ***
## ridageyr     0.1591642  0.0230109   6.917  5.9e-12 ***
## ridreth1     0.0026783  0.3137063   0.009 0.993189    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 18.5 on 2401 degrees of freedom
## Multiple R-squared:  0.1087, Adjusted R-squared:  0.1035 
## F-statistic: 20.91 on 14 and 2401 DF,  p-value: < 2.2e-16
rmse(model.glucuse, test.data.glucose)
## [1] 16.59664

I then fitted a support vector model to compare to the other two models to see which model performed the best.

set.seed (100)
svm.glucose<-svm(lbxsgl ~ ., data=train.data.glucose, kernel="linear", cost=1, scale=TRUE)
print(svm.glucose)
## 
## Call:
## svm(formula = lbxsgl ~ ., data = train.data.glucose, kernel = "linear", 
##     cost = 1, scale = TRUE)
## 
## 
## Parameters:
##    SVM-Type:  eps-regression 
##  SVM-Kernel:  linear 
##        cost:  1 
##       gamma:  0.07142857 
##     epsilon:  0.1 
## 
## 
## Number of Support Vectors:  2005
svm.pred<-predict(svm.glucose, newdata=train.data.glucose)
table(svm.pred, train.data.glucose$lbxsgl)
##                   
## svm.pred           51 55 56 57 58 60 61 63 64 65 66 67 68 69 70 71 72 73
##   80.0562078116171  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   80.3913596110989  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   80.5139042665649  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   81.0510909137322  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1
##   81.0707415155938  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   81.2098924661798  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   81.3323008567811  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   81.3406131814013  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   81.4341649480316  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   81.4380653100961  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   81.480033115421   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   81.4874199790237  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   81.48998537045    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   81.5208310716737  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   81.6808033618252  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   81.6965411229183  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   81.715866655641   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   81.7511142855645  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   81.7631148456107  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   81.7777055319578  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   81.9061123313775  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   82.0292414220675  0  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   82.0797427415111  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   82.1268436374399  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  0
##   82.1423449459458  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   82.1954875315059  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   82.2397793259966  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   82.244201944717   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   82.2630172741124  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   82.2691704231734  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   82.3230963457374  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   82.3274174853165  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   82.3790173536243  0  0  0  0  0  0  0  0  0  1  0  0  0  0  0  0  0  0
##   82.3902664624284  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
##   82.4063762466212  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  0
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##   89.3140833771628   0   0   0   0   0   0   0   0   0   0   0
##  [ reached getOption("max.print") -- omitted 1576 rows ]
rmse(svm.glucose, test.data.glucose)
## [1] 16.78708
features<-x.train
outcome<-train.data.glucose$lbxsgl

svm_tune <- tune(svm, train.x=features, train.y=outcome,  kernel="linear", range=list(cost=10^(-1:2)))

summary(svm_tune)
## 
## Parameter tuning of 'svm':
## 
## - sampling method: 10-fold cross validation 
## 
## - best parameters:
##  cost
##   100
## 
## - best performance: 352.6775 
## 
## - Detailed performance results:
##    cost    error dispersion
## 1   0.1 353.0315   141.0055
## 2   1.0 352.8538   140.7250
## 3  10.0 352.7076   140.6454
## 4 100.0 352.6775   140.6988
svm.glucose.new<-svm(lbxsgl ~ ., data=train.data.glucose, kernel="linear", cost= 100, scale=TRUE)

print(svm.glucose.new)
## 
## Call:
## svm(formula = lbxsgl ~ ., data = train.data.glucose, kernel = "linear", 
##     cost = 100, scale = TRUE)
## 
## 
## Parameters:
##    SVM-Type:  eps-regression 
##  SVM-Kernel:  linear 
##        cost:  100 
##       gamma:  0.07142857 
##     epsilon:  0.1 
## 
## 
## Number of Support Vectors:  2006
svm.pred.new<-predict(svm.glucose.new, newdata=test.data.glucose)
table(svm.pred.new, test.data.glucose$lbxsgl)
##                   
## svm.pred.new       56 57 58 59 60 62 64 65 66 67 68 69 70 71 72 73 74 75
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##   98.4838514752096   0   0   0   0
##   98.5225722482995   0   0   0   0
##   98.5899471267307   0   0   0   0
##   98.6016687093249   0   0   0   0
##   98.6486612292774   0   0   0   0
##   98.6551802407761   0   0   0   0
##   98.7183604463985   0   0   0   0
##   98.7206573231248   0   0   0   0
##   98.7223932205121   0   0   0   0
##   98.7558200026583   0   0   0   0
##   98.7705783087456   0   0   0   0
##   98.8021589404983   0   0   0   0
##   98.9046625579936   0   0   0   0
##   98.9475178557835   0   0   0   0
##   98.9566998177178   0   0   0   0
##   98.9587457332099   0   0   0   0
##   98.9762758950698   0   0   0   0
##   99.0183336710567   0   0   0   0
##   99.0211428927197   0   0   0   0
##   99.040764803145    0   0   0   0
##   99.1649930414009   0   0   0   0
##   99.1885787698014   0   0   0   0
##   99.2700154441043   0   0   0   0
##   99.3219685002108   0   0   0   0
##   99.3233711297254   0   0   0   0
##   99.3704939608842   0   0   0   0
##   99.714754573137    0   0   0   0
##   99.7370371072551   0   0   0   0
##   99.7422856069464   0   0   0   0
##   99.7711386733129   0   0   0   0
##   99.7776326672318   0   0   0   0
##   99.8223378762241   0   0   0   0
##   99.8482779740638   0   0   0   0
##   99.8697035167904   0   0   0   0
##   100.00350315725    0   0   0   0
##   100.025914075392   0   0   0   0
##   100.038810931818   0   0   0   0
##   100.159341493571   0   0   0   0
##   100.332069846762   0   0   0   0
##   100.477626759249   0   0   0   0
##   100.562432940703   0   0   0   0
##   100.580714559439   0   0   0   0
##   100.692244300054   0   0   0   0
##   100.698112898322   0   0   0   0
##   100.805519068685   0   0   0   0
##   100.807143169366   0   0   0   0
##   100.933605134299   0   0   0   0
##   101.050660256328   0   0   0   0
##   101.215651052697   0   0   0   0
##   101.252735616621   0   0   0   0
##   101.562460160282   0   0   0   0
##   101.60698065388    0   0   0   0
##   101.655570353753   0   0   0   0
##   101.76996269079    0   0   0   0
##   101.821555288001   0   0   0   0
##   101.879403374115   0   0   0   0
##   102.378677608246   0   0   0   0
##   102.607368484064   0   0   0   0
##   102.920319435367   0   0   0   0
##   103.188708645053   0   0   0   0
##   103.52135924736    0   0   0   0
##  [ reached getOption("max.print") -- omitted 3 rows ]
rmse(svm.glucose.new, test.data.glucose)
## [1] 16.79641

The support vector machine used the largest cost of 100 that shows there is a lot of misclassiofcation in in this model.

Part 3: Ethical considerations of data-driven analyses in social epidemiology

The following is an excerpt from an article on the ethical tensions in using social media data to characterize individuals’ and communities’ mental health. After reading the brief excerpt, address both questions listed below. The response (to both questions) should be limited to one page.

“Powered by machine learning techniques, social media provides an unobtrusive lens into individual behaviors,emotions, and psychological states. Recent research has successfully employed social media data to predictmental health states of individuals, ranging from the presence and severity of mental disorders like depression to the risk of suicide. These algorithmic inferences hold great potential in supporting early detection and treatment of mental disorders and in the design of interventions. At the same time, the outcomes of this research can pose great risk…”

Question 1: Describe one potential risk to either individuals, communities or specific populations that could arise from research or public health practice that utilizes data-driven analyses of social-media data.

Question 2: Describe one potential safeguard that could be implemented to prevent the risk you describe.